Probabilistic Methods in Combinatorics - 0366.4913.01

Noga Alon ( nogaa@tau.ac.il )
Spring 2016-2017, Tuesday 16-19, Schreiber 6
School of Mathematical Sciences,
Tel-Aviv University

Procedural Matters:

Prerequisite Courses: Discrete Mathematics, Introduction to Probability.
Exercises will be given during the course and their solutions will be graded.

Text books:

Most of the topics covered in the course appear in the books listed below (especially the first one). Other topics appear in recent papers, many of which can be found in the journal Random Structures and Algorithms.
N. Alon and J. H. Spencer, The Probabilistic Method, Wiley, 1992. (Second Edition, 2000, Third Edition 2008, Fourth Edition 2016).
B. Bollobas, Random Graphs, Academic Press, 1985. (Second Edition, Cambridge University Press, 2001.)
S. Janson, T. Luczak and A. Rucinski, Random Graphs, Wiley, 2000.
M. Molloy and B. Reed, Graph Colouring and the Probabilistic Method, Springer, 2002.

Course syllabus:

Probabilistic methods in Combinatorics and their applications in theoretical Computer Science. The topics include linearity of expectation, the second moment method, the local lemma, correlation inequalities, martingales, large deviation inequalities, geometry, derandomization.

Course Outline (to be updated during the term):

  1. : March 14
    The basic method, Ramsey numbers, Dominating sets in graphs, Hypergraph 2-coloring, Sum-free subsets, Set pairs theorem.
  2. : March 21
    Linearity of expectation, Hamiltonian paths in tournaments and the conjecture of Szele, Minc Conjecture.
  3. : March 28
    No class
  4. : April 4
    The second moment method, Turan's proof of the Hardy Ramanujan theorem, distinct sums, random graphs and threshold functions, cliques in random graphs.
  5. : April 5
    More on cliques in random graphs. Alterations: Graphs with high girth and high chromatic number, bounding of large deviations and consistent arcs in tournaments.
  6. : April 11,18
    Passover
  7. : April 25
    The local lemma: the general lemma and its symmetric version, Straus' problem, directed cycles.
  8. : May 2
    Independence Day
  9. : May 9
    Correlation inequalities: the four functions theorem and its applications, the FKG Inequality, The Harris-Kleitman Theorem, correlation between properties of random graphs.
  10. : May 16
    Martingales: the edge exposure and the vertex exposure martingales, Azuma's Inequality.
  11. : May 23
    More martingales, chromatic number of sparse random graphs, solution of exercises.
  12. : May 30
    Shavuoth
  13. : June 6
    Poisson approximation: The Janson Inequalities and their application for constructing Ramsey type graphs.
  14. : June 13
    The chromatic number of G(n,1/2). Geometry: the VC dimension of a range space and its applications.
  15. : June 20
    More on the VC dimension and its applications. Dominating sets in k-majority tournaments.
  16. : June 27
    Gems including crossing numbers, incidences and additive number theory, the Ramsey number r(3,k), summary.

Exercises